# coneOfLines -- cone of lines on a subvariety passing through a point

## Synopsis

• Usage:
coneOfLines(X,p)
coneOfLines(p,X)
• Inputs:
• X, , a subvariety of $\mathbb{P}^n$
• p, , a point on $X$
• Outputs:
• , the subscheme of $\mathbb{P}^n$ consisting of the union of all lines contained in $X$ and passing through $p$

## Description

In the example below we compute the cone of lines passing through the generic point of a smooth del Pezzo fourfold in $\mathbb{P}^7$.

 i1 : K := frac(QQ[a,b,c,d,e]); t = gens ring PP_K^4; phi = rationalMap {minors(2,matrix{{t_0,t_1,t_2},{t_1,t_2,t_3}}) + t_4}; o3 : MultirationalMap (rational map from PP^4 to PP^7) i4 : X = image phi; o4 : ProjectiveVariety, 4-dimensional subvariety of PP^7 i5 : ideal X 2 2 o5 = ideal (t - t t + t t , t t - t t + t t , t - t t + t t , t t - 5 4 6 2 7 4 5 3 6 1 7 4 3 5 0 7 2 4 ------------------------------------------------------------------------ t t + t t , t t - t t + t t ) 1 5 0 6 2 3 1 4 0 5 o5 : Ideal of frac(QQ[a..e])[t ..t ] 0 7 i6 : p := projectiveVariety minors(2,(vars K)||(vars ring PP_K^4)) o6 = point of coordinates [a/e, b/e, c/e, d/e, 1] o6 : ProjectiveVariety, a point in PP^4 i7 : coneOfLines(X,phi p) o7 = surface in PP^7 cut out by 6 hypersurfaces of degrees 1^3 2^3 o7 : ProjectiveVariety, surface in PP^7 i8 : ideal oo 2 -d 2c -b - c + b*d -d c b o8 = ideal (t + --t + --t + --t + ----------t , t + --t + -t + -t + 2 e 4 e 5 e 6 2 7 1 e 3 e 4 e 5 e ------------------------------------------------------------------------ 2 -a - b*c + a*d -c 2b -a - b + a*c 2 --t + -----------t , t + --t + --t + --t + ----------t , t - t t e 6 2 7 0 e 3 e 4 e 5 2 7 5 4 6 e e ------------------------------------------------------------------------ 2 d -2c b c - b*d 2 d -c + -t t + ---t t + -t t + --------t , t t - t t + -t t + --t t + e 4 7 e 5 7 e 6 7 2 7 4 5 3 6 e 3 7 e 4 7 e ------------------------------------------------------------------------ -b a b*c - a*d 2 2 c -2b a --t t + -t t + ---------t , t - t t + -t t + ---t t + -t t + e 5 7 e 6 7 2 7 4 3 5 e 3 7 e 4 7 e 5 7 e ------------------------------------------------------------------------ 2 b - a*c 2 --------t ) 2 7 e o8 : Ideal of frac(QQ[a..e])[t ..t ] 0 7